Cholesky Decomposition

Cholesky decomposition is a special kind of matrix matrix decomposition of the Lower-Upper English, which consists in the factorization of a matrix in the product of two or more matrices.

In other words, Cholesky decomposition consists of matching a matrix containing the same number of rows and columns (square matrix) to a matrix with zeros above the main diagonal multiplied by its matrix transposed with zeros below the main diagonal .

LU decomposition, unlike Cholesky, can be applied to various types of square matrices.

Cholesky decomposition characteristics

Cholesky decomposition consists of:

  • An upper triangular square matrix: A squarematrix that only has zeros below the main diagonal.
  • A lower triangular square matrix:Matrix that only has zeros above the main diagonal.

Mathematically, if there is a positive definite symmetric matrix, E , then there is a lower triangular symmetric matrix, K, of the same dimension as E , resulting in:

The previous matrix appears as the Cholesky matrix of E. This matrix acts as the square root of the matrix E. We know that the domain of the square root is:

{X : x ≥ 0}

Which is defined in all non-negative real numbers. In the same way as the square root, Cholesky’s matrix will only exist if the matrix is ​​defined semi-positive. A matrix is ​​defined semi-positive when the minor principals have a positive or zero determinant.

Cholesky decomposition of E is a diagonal matrix such that:

We can see that the matrices are square and contain the mentioned characteristics; triangle of zeros above the main diagonal in the first matrix and triangle of zeros below the main diagonal in the transformed matrix.

Applications of Cholesky decomposition

In finance it is used to transform the realizations of independent normal variables into correlated normal variables according to an E correlation matrix .

If N is an independent vector normal (0.1), it holds that N is a vector Standard (0.1) correlated as E .

Example of Cholesky decomposition

This is the simplest example we can find of Cholesky decomposition since the matrices have to be square, in this case, the matrix is ​​(2 × 2). Two rows by two columns. In addition, it meets the characteristics of having zeros above and below the main diagonal. This matrix is ​​defined semi-positive because the minor principals have a positive determinant. We define:

Solving for: c 2 = 4; b · c = -2; a 2 + b 2 = 5; We have four possible Cholesky matrices:

 

Finally we calculate to find (a, b, c). Once we find them, we will have Cholesky matrices. The calculation is as follows:

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